Left-exact Localizations of $\infty$-Topoi III: The Acyclic Product
Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal
TL;DR
The paper develops a conceptual framework linking Goodwillie calculus in the unstable topos-theoretic setting to towers of left-exact localizations via a derived acyclic product. It constructs a symmetric monoidal closed structure on left-exact localizations (the acyclic product) and proves congruences are closed under this product, enabling completion towers that mirror I-adic towers in commutative algebra. It introduces nilradical, hyper-radical, hyper-reduction, and separation to organize infinitesimal and connected data, and shows Goodwillie towers arise as completion towers for the free logos on one generator, classifying nilpotent objects and expounding pointed variants. The framework yields a robust Logos–Topos dictionary, providing tools such as decalage and suspension to control operations across modes (monogenic, epic, and hyper-radical aspects) and to reinterpret n-excisive functors as classifying nilpotent objects. Collectively, this work reframes unstable homotopy calculus in a principled topos-theoretic language, enabling new algebraic techniques for understanding strata of excisiveness and completion in higher topos theory.
Abstract
We define a commutative monoid structure on the poset of left-exact localizations of a higher topos, that we call the acyclic product. Our approach is anchored in a structural analogy between the poset of left-exact localizations of a topos and the poset of ideals of a commutative ring. The acyclic product is analogous to the product of ideals. The sequence of powers of a given left-exact localization defines a tower of localizations. We show how this recovers the towers of Goodwillie calculus in the unstable homotopical setting. We use this to describe the topoi of $n$-excisive functors as classifying $n$-nilpotent objects.